A327438 -id:A327438 - cp's OEIS frontend

A327352 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

1, 1, 1, 4, 1, 14, 4, 1, 83, 59, 23, 2, 1232, 2551, 2792, 887, 107, 10, 1

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Triangle begins:
     1
     1    1
     4    1
    14    4    1
    83   59   23    2
  1232 2551 2792  887  107   10    1
Row n = 3 counts the following antichains:
  {}             {{1,2,3}}      {{1,2},{1,3},{2,3}}
  {{1}}          {{1,2},{1,3}}
  {{2}}          {{1,2},{2,3}}
  {{3}}          {{1,3},{2,3}}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1},{2}}
  {{1},{3}}
  {{2},{3}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{2},{3}}

Row sums are A014466.
Column k = 0 is A327355.
The unlabeled version is A327438.
Cf. A052446, A327062, A327071, A327103, A327111, A327144, A327351, A327353.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],spanEdgeConn[Range[n],#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

Original entry on oeis.org

1, 1, 3, 6, 15, 52, 410, 32697

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichains:
  {}  {}         {}             {}
      {{1}}      {{1}}          {{1}}
      {{1},{2}}  {{1,2}}        {{1,2}}
                 {{1},{2}}      {{1},{2}}
                 {{1},{2,3}}    {{1,2,3}}
                 {{1},{2},{3}}  {{1},{2,3}}
                                {{1,2},{1,3}}
                                {{1},{2},{3}}
                                {{1},{2,3,4}}
                                {{1,2},{3,4}}
                                {{1},{2},{3,4}}
                                {{1},{2},{3},{4}}
                                {{2},{1,3},{1,4}}
                                {{1,2},{1,3},{2,3}}
                                {{4},{1,2},{1,3},{2,3}}

Column k = 0 of A327438.
The labeled version is A327355.
The covering case is A327426.
Cf. A014466 A052446, A120338, A261005, A293606, A327201, A327236, A327352, A327353, A327354, A327438.

a(n > 0) = A306505(n) - A261006(n).

cp's OEIS Frontend

A327352 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

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